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National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
2023 Spain Mathematical Olympiad
2
2
Part of
2023 Spain Mathematical Olympiad
Problems
(1)
Equal lengths on altitude
Source: Spain MO 2023 P2
3/10/2023
Let
A
B
C
ABC
A
BC
be an acute scalene triangle with incenter
I
I
I
and orthocenter
H
H
H
. Let
M
M
M
be the midpoint of
A
B
AB
A
B
. On the line
A
H
AH
A
H
we consider points
D
D
D
and
E
E
E
, such that the line
M
D
MD
M
D
is parallel to
C
I
CI
C
I
and
M
E
ME
ME
is perpendicular to
C
I
CI
C
I
. Prove that
A
E
=
D
H
AE=DH
A
E
=
DH
.
geometry